# ICME GRP1 HW3&4

## Contents

### Homework 3 & 4 Contributions

#### FEA Overview

Finite element analysis (FEA) is a numerical technique based on the method of weighted residuals used for approximating the solutions of boundary value problems across domains whose geometry may be of varying complexity. In FEA, this is done by taking advantage of the fundamental theorem of variational calculus, which forces the approximate solution of the boundary value problem to be equal to actual solution at specific points in the domain, hereafter referred to as nodes. These nodes define the boundaries of subdomains, known as elements, across which, the approximate solution is interpolated. In this way, a smooth and continuous approximate solution is constructed, one which is guaranteed to converge to the actual solution at the nodes and the boundaries of the domain.

##### Galerkin Method of Weighted Residuals

FEA is based on the method of weighted residuals. This method makes use of a space of functions, known as trial functions, whose values satisfy the prescribed boundary conditions. Given a differential equation of the form

with boundary conditions of the form

the approximate solution, y* may be assumed to be of the form

where c and N are unknown coefficients and trial functions, respectively, corresponding to the “ith” of n elements. In the above example, the summation of c and N terms represents the approximate solution of the homogeneous part of D, whereas h(x) represents the heterogeneous part.

Upon substituting this approximate solution for the real solution in D, it is a residual error, R(x), may be found. At this point, the integral of this weighted form of the residual, which is assumed to take the value of zero at each node, may be evaluated in order to solve for the constants, ci. In the Galerkin method, the weight functions are assumed to be identical to the trial functions, thus

Given that the weight functions are assumed to be non-zero at the nodes, the fundamental theorem of variational calculus forces the value of the residual to zero, thereby guaranteeing that the approximate solution converges to the actual solution at the nodes.

##### FEA

In FEA, the unknown constants ci are equated with the unknown values, yi, of the solution function at each node. When this system of integrals is evaluated, it results and a system of n algebraic equations which may be combined into the stiffness matrix K. At this point, when combined with the vector of boundary conditions F, the classic matrix/vector equation

may be constructed and solved for the values of y.